Continuum mechanics

In continuum mechanics, instead of dealing with discrete particles with individual masses $m_{i}$ , we work with a continuous distribution of mass described by a mass density field $ρ (x)$ (is not the same as Classical Field Theory?). This field gives the mass per unit volume at each point in space. The total mass $M$ in a given region $V$ of space is then found by integrating the mass density over that region:

M = \int_{V} ρ (x) d V .

To recover the idea of an isolated mass $m$ located at a specific position $x^{'}$ , the mass density can be described using the Dirac delta function:

ρ (x) = m δ (x - x^{'}) .

This equation means that the mass is concentrated entirely at the point $x^{'}$ . The total mass can be recovered by integrating $ρ (x)$ over the entire space:

M = \int_{all space} ρ (x) d V = \int_{all space} m δ (x - x^{'}) d V = m .